DEPARTING B722 CREW RESPONDED TO A PREVENTIVE RA FROM TCASII.

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WE WERE ON DEP FROM DFW ENRTE TO BUR. WE WERE AT 10000 FT MSL; LEVEL FLT; ON A 260 DEG HDG; ON VECTORS. WE WERE WORKING WITH REGIONAL DEP CTL ON 119.4. ATC POINTED OUT TFC AT OUR 2 O'CLOCK POS AND 11000 FT. I (THE FO) RESPONDED BY SAYING 'TFC IN SIGHT.' AFTER A FEW SECONDS; THE CTLR ASKED IF WE COULD MAINTAIN 'VISUAL' ON THAT ACFT AND I RESPONDED 'YES.' THEN IMMEDIATELY THE CTLR DIRECTED US TO 'TURN TO A HDG OF 280 DEGS; MAINTAIN VISUAL SEPARATION WITH THAT TFC; AND CLB AT YOUR DISCRETION TO 17000 FT.' I READ BACK THE INSTRUCTIONS EXACTLY AS THEY WERE GIVEN AND THEN LOOKED AND WATCHED THE TARGET TO GAUGE OUR SEPARATION. I DIDN'T FEEL IT WAS THE BEST OF CONDITIONS TO JUDGE SEPARATION OR CLOSURE; SO I REFERRED TO THE TCASII TO SEE IF I COULD GET SOME RANGE AND ALT INFO. IT WAS AT THAT TIME I SAW TFC AT +800 FT AND I TOLD THE CAPT 'MAYBE WE SHOULD JUST STAY AT 10000 FT FOR NOW.' I NOTICED HE HAD STARTED A GRADUAL CLB THOUGH; SO I BEGAN READING OUT OUR TFC'S ALT IN FT ABOVE OURS. THE CAPT HAD THEM LEVELED OFF AND EVEN TRIED TO GO BACK DOWN AS FAST AS THE AUTOPLT COULD TRANSITION. TCASII AUDIO NOW HAD GONE THROUGH ITS 'TFC' PHASE AND NOW WAS SAYING 'MONITOR VERT SPD.' AT THAT TIME WE ALSO HEARD THE OTHER ACFT TELL ATC THEY WERE ON AN RA CLB. I DO NOT RECALL WHAT OUR HORIZ SEPARATION WAS. THE OTHER AFT WAS AN MD80; APPARENTLY FLYING IN ON THE BOWIE STAR TOWARD DFW. AT THE TIME; WE WERE THE ONLY 2 ACFT IN THE ENTIRE NW QUADRANT OF THE DFW AREA. WHILE I VIEW IT AS OUR MISTAKE TO ACCEPT A VISUAL SEPARATION CLRNC UNDER THIS SIT AND OUR MISTAKE FOR DEPARTING 10000 FT MSL LEVEL FLT; I CANNOT IMAGINE WHY WE WERE GIVEN A VECTOR WHICH WOULD BRING OUR FLT PATHS EVEN CLOSER. THIS SERVED TO REDUCE VISUAL CLUES WHILE INTRODUCING XING FLT PATHS AND ALTS. WE TOOK THE BAIT AND THIS SHOULD SERVE AS A LESSON TO BE NOT SO QUICK TO TAKE A CLRNC SUCH AS THIS.